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2. Consider a homogeneous differential equation, p(т, у) dx + q(z, у) dy %3D 0, where the coefficients p(x, y) and q(x, y) satisfy the homogeneity con- dition: p(Аг, Ау) — А"р(a, у); q(λ, λ) - λq(α, 9). Explain why it is always possible to express any such homogeneous differential equation in the form dy = F dx () HINT: You could start by first proving that p(x, y) = x°p(1,y/x), and q(x, y) = x°q(1,y/x).

2. Consider a homogeneous differential equation, p(т, у) dx + q(z, у) dy %3D 0, where the coefficients p(x, y) and q(x, y) satisfy the homogeneity con- dition: p(Аг, Ау) — А"р(a, у); q(λ, λ) - λq(α, 9). Explain why it is always possible to express any such homogeneous differential equation in the form dy = F dx () HINT: You could start by first proving that p(x, y) = x°p(1,y/x), and q(x, y) = x°q(1,y/x).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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2. Consider a homogeneous differential equation, p(х, у) da + q(т, у) dy %3D 0, where the coefficients p(x, y) and g(x, y) satisfy the homogeneity con- dition: p(Ax, \y) = X®p(x, Y); α(λ, λ9) -λάq(π, y). Explain why it is always possible to express any such homogeneous differential equation in the form dy = F dx HINT: You could start by first proving that p(x, y) = x°p(1, y/x), and q(x, y) = xªq(1,y/x).
Transcribed Image Text:2. Consider a homogeneous differential equation, p(х, у) da + q(т, у) dy %3D 0, where the coefficients p(x, y) and g(x, y) satisfy the homogeneity con- dition: p(Ax, \y) = X®p(x, Y); α(λ, λ9) -λάq(π, y). Explain why it is always possible to express any such homogeneous differential equation in the form dy = F dx HINT: You could start by first proving that p(x, y) = x°p(1, y/x), and q(x, y) = xªq(1,y/x).
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